MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

64 Robert Goldblattatomic commands by the operations of alternation, composition and iteration. AKripke model for PDL assigns a binary relation to each atomic program, and theninterprets complex programs by the above conditions on R π∪π ′, R π;π ′ and R π ∗.Fischer and Ladner proved that this semantically defined logic has the finite modelproperty by a version of the filtration construction. That method produces a falsifyingmodel for a given non-theorem α whose size is exponential in the length of α.The result was used to establish an upper bound of nondeterministic exponentialtime for the complexity of the satisfiability problem: there is a nondeterministicalgorithm for deciding PDL-satisfiability that runs in a time bounded above byan exponential function c n of the length n of the formula concerned (for someconstant c). They also gave a lower bound of deterministic exponential time forthe complexity of this problem: there is a constant d > 1 such that no deterministicalgorithm can decide the satisfiability question for all formulas in timeless than d n . The technique used was to construct a PDL-formula that encodesthe computations of a certain kind of Turing machine that was known to requireexponential running time. The gap between these upper and lower bounds wasclosed by Pratt [1980b], who used Hintikka’s model sets and tableaux methods togive a deterministic exponential time algorithm for deciding satisfiability/validityin PDL.A finite axiomatisation of PDL was proposed in [Segerberg, 1977], the mostnotable feature being the induction axiomp → ([π ∗ ](p → [π]p) → [π ∗ ]p).The first proof of completeness for PDL was published by Rohit Parikh [1978a],with other proofs being attributed to Gabbay, Segerberg [1982] and Pratt. 58 Thefirst extensive study of quantificational dynamic logic was made in David Harel’s1978 dissertation under Pratt’s supervision, published as [Harel, 1979].Many variants of dynamic logic have been studied by varying the modelling,the set of formulas, and the set of programs having associated modalities. Deterministicprograms are modelled by requiring R π to be a functional relation.Program predicates may be used to express computational behaviour of particularprograms, such as loop(π), meaning that some execution of π fails to terminate,and repeat(π), meaning that π can be repeatedly executed infinitely many times.PDL programs can be viewed as regular sets of sequences of basic commands, butallowing context-free sets of sequences as programs results in a stronger logic thatis Π 1 1-complete and hence highly undecidable. This was shown by Harel, Pnueliand Stavi [1983].Dynamic algebras were introduced by Dexter Kozen and Pratt in 1979 and theirstructure and representations investigated in a number of papers. 59 They comprisea “Kleene algebra” that abstracts the algebra of regular expressions and acts asa collection of operators on a Boolean algebra. Concrete models are provided by58 More background on the beginnings of dynamic logic is provided in [Goldblatt, 1986].59 See [Kozen and Tiuryn, 1990] for references.